GLOBAL C1 REGULARITY OF THE VALUE FUNCTION IN OPTIMAL STOPPING PROBLEMS

• 成果类型: Article
• 署名作者: De Angelis, T.; Peskir, G.
• 署名单位: University of Leeds; University of Manchester
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1517
• 发表日期: 2020
• 页码: 1007-1031
• 关键词: of-variable formula; harmonic-functions; dirichlet problem; boundary points; Local Time; continuity

摘要

We show that if either the process is strong Feller and the boundary point is probabilistically regular for the stopping set, or the process is strong Markov and the boundary point is probabilistically regular for the interior of the stopping set, then the boundary point is Green regular for the stopping set. Combining this implication with the existence of a continuously differentiable flow of the process we show that the value function is continuously differentiable at the optimal stopping boundary whenever the gain function is so. The derived fact holds both in the parabolic and elliptic case of the boundary value problem under the sole hypothesis of probabilistic regularity of the optimal stopping boundary, thus improving upon known analytic results in the PDE literature, and establishing the fact for the first time in the case of integro-differential equations. The method of proof is purely probabilistic and conceptually simple. Examples of application include the first known probabilistic proof of the fact that the time derivative of the value function in the American put problem is continuous across the optimal stopping boundary.